Twelve traders have distinct annual P&Ls whose joint distribution is exchangeable, with no independence assumed. Reading them in a uniformly random order, compute the expected number who set a new running maximum, and use the same argument for the probability that trader 1 finishes in the top three.

Twelve traders have distinct annual P&Ls whose joint distribution is exchangeable, with no independence assumed. Reading them in a uniformly random order, compute the expected number who set a new running maximum, and use the same argument for the probability that trader 1 finishes in the top three.

Approach: Define an indicator for each position being a new record, find its probability from symmetry alone, and sum without needing any independence.

3.1032. Exchangeability means every ordering of the twelve values is equally likely, which is all the argument uses. The k-th value read sets a new running maximum exactly when it is the largest of the first k, and by symmetry that has probability 1/k whatever the dependence between the underlying P&Ls. Summing the indicators and applying linearity of expectation, which never requires independence, gives the harmonic number H_12 = 1 + 1/2 + ... + 1/12 = 3.1032. The same symmetry gives each trader an equal chance of every rank, so P(trader 1 in the top three) = 3/12 = 1/4. The expected rank is 6.5. Records grow like log(n) + 0.5772, so doubling the desk to 24 traders raises the expected count only to 3.78. A firm reporting three record years in twelve is therefore reporting the base rate, and treating that run as evidence of anything is a mistake the arithmetic settles.

Follow-up: What is the variance of the number of records among n exchangeable values, and why does the answer still not need independence?

Key concepts: exchangeability, linearity of expectation, harmonic number, records.